Theorem hmphtr 21586

Description: "Is homeomorphic to" is transitive. (Contributed by FL, 9-Mar-2007.) (Revised by Mario Carneiro, 23-Aug-2015.)

Assertion

Ref	Expression
hmphtr	|- ( ( J ~= K /\ K ~= L ) -> J ~= L )

Proof of Theorem hmphtr

Dummy variables f g are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		hmph 21579	. 2 |- ( J ~= K <-> ( J Homeo K ) =/= (/) )
2		hmph 21579	. 2 |- ( K ~= L <-> ( K Homeo L ) =/= (/) )
3		n0 3931	. . 3 |- ( ( J Homeo K ) =/= (/) <-> E. f f e. ( J Homeo K ) )
4		n0 3931	. . 3 |- ( ( K Homeo L ) =/= (/) <-> E. g g e. ( K Homeo L ) )
5		eeanv 2182	. . . 4 |- ( E. f E. g ( f e. ( J Homeo K ) /\ g e. ( K Homeo L ) ) <-> ( E. f f e. ( J Homeo K ) /\ E. g g e. ( K Homeo L ) ) )
6		hmeoco 21575	. . . . . 6 |- ( ( f e. ( J Homeo K ) /\ g e. ( K Homeo L ) ) -> ( g o. f ) e. ( J Homeo L ) )
7		hmphi 21580	. . . . . 6 |- ( ( g o. f ) e. ( J Homeo L ) -> J ~= L )
8	6, 7	syl 17	. . . . 5 |- ( ( f e. ( J Homeo K ) /\ g e. ( K Homeo L ) ) -> J ~= L )
9	8	exlimivv 1860	. . . 4 |- ( E. f E. g ( f e. ( J Homeo K ) /\ g e. ( K Homeo L ) ) -> J ~= L )
10	5, 9	sylbir 225	. . 3 |- ( ( E. f f e. ( J Homeo K ) /\ E. g g e. ( K Homeo L ) ) -> J ~= L )
11	3, 4, 10	syl2anb 496	. 2 |- ( ( ( J Homeo K ) =/= (/) /\ ( K Homeo L ) =/= (/) ) -> J ~= L )
12	1, 2, 11	syl2anb 496	1 |- ( ( J ~= K /\ K ~= L ) -> J ~= L )

Syntax hints:   -> wi 4   /\ wa 384   E.wex 1704   e. wcel 1990   =/= wne 2794   (/)c0 3915   class class class wbr 4653   o. ccom 5118  (class class class)co 6650   Homeochmeo 21556   ~= chmph 21557

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-fv 5896  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-1st 7168  df-2nd 7169  df-1o 7560  df-map 7859  df-top 20699  df-topon 20716  df-cn 21031  df-hmeo 21558  df-hmph 21559

This theorem is referenced by:  hmpher  21587  xrhmph  22746